Frustrated quantum spin systems in small triangular lattices studied with a numerical method

The study of quantum frustrated systems remains one of the most challenging subjects of quantum magnetism, as they can hold quantum spin liquids, whose characterization is quite elusive. The presence of gapped quantum spin liquids possessing long-range entanglement while being locally indistinguishable often demands highly sophisticated numerical approaches for their description. Here we propose an easy computational method based on exact diagonalization with engineered boundary conditions in very small plaquettes. We apply the method to study the quantum phase diagram of diverse antiferromagnetic frustrated Heisenberg models in the triangular lattice. Our results are in qualitative agreement with previous results obtained by means of sophisticated methods like density matrix renormalization group (2D-DMRG) or variational quantum Monte Carlo.

Figure 1

Upper panels: Sketch representation of a quantum ordered Néel phase (left) and a QSL phase (right) in a large lattice. Lower panels: Sketch of twisted boundary conditions in a $4\times 3$ triangular lattice with anisotropic nearest-neighbor interactions. Boundary spins in blue are twisted in the XY plane by an angle ${\varphi }_1$, while red-colored boundary spins are twisted by a phase ${\varphi }_2$. The pink-colored boundary spin (top-left corner) is twisted by an angle $\varphi ={\varphi }_1+{\varphi }_2$. Bottom left: For an ordered 2D-Néel phase along the diagonal directions, the boundary phases which reproduce the order are ${\varphi }_1=0$ and ${\varphi }_2=\pi$. Bottom right: For a quantum disordered phase, such a set of phases cannot be defined. The anisotropy of the SCATL model is depicted by the three different line styles in the bottom panels.

Figure 2

Classical phase diagram for the SCATL for both XY ($\lambda =0$) and Heisenberg ($\lambda =1$) interactions, obtained by plotting $Q_x$ in Eqs. (9) as a function of the anisotropy (top left). The other panels show the spin-structure factor and a sketch of the spin order for each classical phase.

Figure 3

Quantum phase diagram of the SCATL model as obtained through the number of quasidegenerate configurations, $N_c$, for a $4\times 3$ and $4\times 4$ lattice. Upper (lower) panels, XY (Heisenberg) interactions. In the $4\times 4$ XY diagram, the different quantum phases are labeled 1 to 8 (see text for details). The right panels correspond to the quantum phase diagram obtained with MSWT from Ref. [46], where the white regions correspond to the breakdown of the MSWT calculations and are indistinguishably associated with gapless or gapped QSL.

Figure 4

Study of the SCATL quantum phase diagram in the region between two 2D Néel phases. The averaged order parameter, as defined in Eq. (5), is plotted along the vertical white line in Fig. 3 for different lattice sizes. The three arrows indicate the points where the three representative spin-structure factors are plotted.

Figure 5

Quantum phase diagram along the diagonal white line in Fig. 3 for the $4\times 4$ lattice with XY interactions. Top: Averaged structure factor ${\langle S\left(\vec{k}\right)\rangle }_d$. Center: Overlap, $O_p$, versus relative energy, ${\epsilon }_p$. Bottom: Order parameter, $M_p$ (see text). The dashed vertical lines limit the region ${\epsilon }_p<0.01$, where the average is done.

Figure 6

Top: Averaged concurrence between NN sites along the three lattice directions. Bottom: Geometrical entanglement, $E_G$ [Eq. (7)], and projections to separable states (see text). All quantities are averaged over the configurations with ${\epsilon }_p<0.01$, and the dispersion of the values is represented by error bars. The plotted region corresponds to the white diagonal line in Fig. 3, where the values of $t_2$ are chosen accordingly. The arrows indicate the values used in Fig. 5, which are representative of each quantum phase explored.

Figure 7

Energy spectrum of the five lowest states in the $S_z=0$ manifold for a $4\times 4$ lattice in the putative QSL ($t_2=t_3=0.7$) using PBCs (${\varphi }_1={\varphi }_2=0$) as a function of an inserted twisting phase, $\mathrm{\Phi }$, on the boundary along the horizontal direction, simulating an external magnetic field.

Figure 8

Left: Introduced external magnetic flux at which the minimum gap between ground and first excited state is reached in the SCATL-XY model using RTBC in a $4\times 4$ lattice. The flux ${\mathrm{\Phi }}_1$ is implemented by modifying the tunneling coefficients along the $x$ direction as explained in the text. The plotted values correspond to the average over all the compatible configurations $N_c$. Right: Energy spectrum of the five lowest states in the $S_z=0$ manifold as a function of the inserted flux ${\mathrm{\Phi }}_1$ for one single configuration of our RTBC (${\varphi }_1,{\varphi }_2$). The plots correspond to the points indicated with white crosses on the left panel. From bottom to top, (i) 1D Néel ($t_2=t_3=0.1$), (ii) expected gapped QSL ($t_2=t_3=0.7$), and (iii) spiral phase ($t_2=t_3=1$). Clearly, the gap is only closing in the expected gapped QSL phase.

Figure 9

Left: Sketched quantum phase diagram of the $J_1-J_2$ model with chiral interactions obtained from Refs. [27, 28]. Right: $N_c$ for the same model using a $4\times 4$ lattice. The area with large $N_c$ is a signature of a putative gapped QSL phase.

Figure 10

$J_1-J_2$ model without chiral interactions ($J_{\chi }=0$). Upper panel: Number of configurations, $N_c$, with ${\epsilon }_p<0.05$ for different lattice sizes and geometries. First row: $O_p$ and energy bias ${\epsilon }_p$ for a $6\times 4$ lattice. Second row: Our average spin-structure factors, ${\langle S\left(\vec{k}\right)\rangle }_d$. Third row: $S\left(\vec{k}\right)$ obtained with 2D-DMRG taken from Ref. [24].

Figure 11

Sketch of the phase diagram of the hybrid anisotropic $J_1-J_2$ model in the triangular lattice introduced in the text.

Figure 12

Phase diagram of the anisotropic $J_1-J_2$ model [see Eq. (13)] obtained with random twisted boundary conditions for a plaquette $4\times 4$. Our figure of merit is $N_c$, the number of configurations with ground-state energy compatible with the ground state. Top (bottom) panel corresponds to Heisenberg (XY) interactions.

Figure 13

Scheme of twisted boundary conditions in a $4\times 3$ triangular lattice with next-nearest-neighbor interactions (top panel) and chiral interactions (bottom panel). In every interaction term in the periodic boundary, depicted by a black oval, the colored spin is twisted by an angle ${\varphi }_1$ (blue), ${\varphi }_2$ (red) for the left-right and top-bottom boundaries, respectively. Interaction terms which cross two boundaries get twisted by both phases, ${\varphi }_3={\varphi }_1+{\varphi }_2$ (pink) for the left-top boundary, and ${\varphi }_4={\varphi }_1-{\varphi }_2$ (green) for the left-bottom one. The inner bounds are not depicted for clarity.